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  4. Let z be the set of integers and 0 be binary operation of z defined as a 0 b = a + b - ab for all a, b ∈ z. The inverse of an element a( ≠ 1) ∈ z is:

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Q. Let z be the set of integers and 0 be binary operation of z defined as a 0 b = a + b - ab for all a, b $\in$ z. The inverse of an element a( $\neq $1) $\in$ z is:

Relations and Functions - Part 2

Solution:

Let e be the identity element. And we have a .e = a $\Rightarrow $ a + e - ae = a $\Rightarrow $ e - ae = 0 $\Rightarrow $ either e = 0 or 1 - a = 0 but a $\neq$ 1.
Thus e = 0 is the identity.
We know that, $aa^{-1}$ = e;
Now let A is the inverse of a. Thus a . A = 0
$\Rightarrow $ a + A - a A = 0 $\Rightarrow $ A = $\frac{a}{a-1} \in$ Q thus A is the founded inverse of a.